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On 2010 AKHMEDOV and PARK claimed there are infinitely many exotic smooth structures on $S^2\times S^2$, see http://arxiv.org/abs/1005.3346

Then Rasmussen posted a paper : Perfect Morse functions and exotic $S^2 \times S^2$'s http://arxiv.org/abs/1005.4674 But he withdrawn his paper, here is his note:

The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold exists.

Did I understand correctly that the first paper mentioned above had a gap? Which means one don't know whether there exists exotic smooth structure on $S^2\times S^2$.

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  • $\begingroup$ Rasmussen showed that the manifold constructed by AKHMEDOV and PARK has non-vanishing Osvath-Szabo invariant. But the reason he withdrawn is he don't know if it exists. or did I miss something? $\endgroup$ – J. GE Oct 29 '12 at 12:43
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    $\begingroup$ As far as I understand, some experts claim that there is indeed a (serious) gap in this paper related to the computation of fundamental group of the constructed manifold. Since I have not red the paper I don't know if there is a problem there and if there is, what it is. $\endgroup$ – Dmitri Panov Oct 29 '12 at 12:47
  • $\begingroup$ Oszvath-Szabo and Seiberg-Witten invariants are conjecturally equivalent for $4$-manifolds. Their equivalence for $3$-manifolds has been recently established. $\endgroup$ – Liviu Nicolaescu Oct 29 '12 at 13:22
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    $\begingroup$ related closed question:mathoverflow.net/questions/73001/… $\endgroup$ – Ian Agol Oct 29 '12 at 19:06

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